Optimal robust bounds for variance options and asymptotically extreme models

Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January, 2014

Robustness summary Robustness in Option Pricing: make statements about the price of options given very mild modelling assumptions Incorporate market information by supposing the prices of some vanilla options are known Typically expect pricing bounds corresponding to best and worst case models Q: Can we identify these bounds & superhedging strategies? Q: Bounds can be large are there cases where we can get useful bounds? Q: Extremal models are often very unnatural are there cases where normal models are nearly extremal. How to characterise this?

Financial Setting Option priced on an asset S t Dynamics of S t unspecified, but suppose paths are continuous, and we see prices of call options at all strikes K and at maturity time T Assume for simplicity that all prices are discounted this won t affect our main results Under risk-neutral measure, S t should be a (local-)martingale, and we can recover the law of S T at time T from call prices C(K )

Variance Options We may typically suppose a model for (discounted) asset prices of the form: ds t S t = σ t dw t, where W t a Brownian motion. the volatility, σ t, is a locally bounded, progressively measurable process Want to consider options on variance. E.g. a variance swap pays: T ( σt 2 σ 2) dt 0 where σ is the strike. Dupire (1993) and Neuberger (1994) gave a simple replication strategy for such an option. (More recently, Davis-Obłój-Ramal, 2013).

Hedge of Variance Swap Dupire s hedge: Itô implies d (ln S t ) = σ t dw t 1 2 σ2 t dt Hold portfolio short 2 contracts paying ln(s T ), long 2/S t units of asset At time T, portfolio will be worth (up to constant) T 0 σ2 t dt Note that the only modelling assumption here is that the volatility process exists! Note also that ln S T = T 0 σ2 t dt, where t is quadratic variation. Observe the form of this identity: Payoff = Martingale + H(S T ).

Variance Options A variance call is an option paying: ( ln S T K ) + More general options of the form: F( ln S T ). E.g.: volatility swap, payoff: ln S T K. Also payoffs dependent on weighted realised variance: T RVT λ = λ(s t ) d ln S t = 0 T 0 λ(s t )σ 2 t dt. E.g.: options on corridor variance or a gamma swap: T 0 1 {St [a,b]} d ln S t, T 0 S t d ln S t.

Options on (weighted) realised variance Let λ(x) be a strictly positive, continuous function, τ t := RV λ T = t 0 λ(s s)σ 2 s ds and A t such that τ At = t. Then W t = A t 0 σ sλ(s s ) 1/2 dw s is a BM w.r.t. Ft = F At, and if we set X t = S At, we have: d X t = X t λ( X t ) 1/2 d W t = σ( X t ) d W t. Xt is now a diffusion on natural scale ( X τt, τ T ) = (S T, RV λ T ) Knowledge of L(S T ) = L( X τt ).

Variance Call This suggests finding lower/upper bound on price of variance call (say) with given call prices is equivalent to: minimise/maximise: E(τ K ) + subject to: L( X τ ) = µ where µ is a given law. The problem of finding a stopping time τ such that X τ µ is known as the Skorokhod Embedding problem. Q: Are there Skorokhod Embeddings which do this?

B R R + a barrier if: (x, t) B = (x, s) B for all s t Given µ, and D = B C, exists stopping time τ D = inf{t 0 : ( X t, t) D} Root s Construction which is an embedding. Minimises E(τ K ) + over all (UI) embeddings Root (1969) Rost (1976) C. & Wang (2013) Oberhauser & dos Reis (2013)

B R R + a reversed barrier if: (x, t) B = (x, s) B for all s t Given µ, and D = B C, exists a stopping time Rost s Construction τ D = inf{t 0 : ( X t, t) D} which is an embedding. Minimises E(τ K ) + over all (UI) embeddings Rost (1971) Chacon (1985) McConnell (1991) C. & Peskir (2012)

Variance Call Finding bound on price of variance call with given call prices is equivalent to: min/maximise: E(τ K ) + subject to: L( X τ ) = µ where µ is a given law. These are (essentially) the problems solved by Root s and Rost s Barriers! Robustness via Skorokhod Embedding dates back to Hobson ( 98). More recently important generalisations as martingale optimal transport. (Galichon, Henry-Labordère & Touzi ( 11); Beiglböck, Henry-Labordère & Penkner; Tan & Touzi; Dolinsky & Soner,... ( 13)) The connection to Variance options has been observed by a number of authors: Dupire ( 05), Carr & Lee ( 09), Hobson ( 09).

Questions Question This known connection leads to two important questions: 1. How do we find the Root/Rost stopping times? 2. Is there a corresponding hedging strategy? Dupire gave a connected free boundary problem for Root. In C. & Wang, gave a variational characterisation of Root s barrier & construction of optimal strategy; Oberhauser & dos Reis gave characterisation as viscosity solution. Key step in construction for Root: classical results on existence of solution.

Computing Rost s Barrier Suppose I = (0, ), λ C 1 (I) D and, λ(x) 1 and λ (x)λ(x) 2 x are bounded on (0, ). Recall σ(x) = xλ(x) 1/2, and write U µ (x) = x y µ(dy). Theorem Suppose D is Rost s reversed barrier. Then u(x, t) = U µ (x) + E ν x Xt τd S is the unique bounded viscosity solution to: ( u σ(x) 2 2 ) u (x, t) = (x, t) t 2 x 2 + u(0, x) = U µ (x) U δ0 (x). Moreover, given a solution u, a reversed barrier D which solves the SEP can be recovered by D = {(x, t) : u(x, t) > u(0, t)}. See also Oberhauser & dos Reis (2013).

Computing the Barrier 0-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8 0 1 2 3 4 5 6 7 8 9 10 Optimal stopping interpretation: ] u(x, t) = sup τ t E [U x µ ( X τ ) U δ0 ( X τ )

Optimality of Rost s Barrier Chacon s Result Given a function F which is convex, increasing, Rost s (reversed) barrier solves: Want: A simple proof of this... maximise EF( X τ ) subject to: Xτ µ τ a stopping time... that identifies a financially meaningful hedging strategy. For simplicity, consider the case where S 0 is non-random.

Write f (t) = F (t), define Optimality M(x, t) = E (x,t) f (τ D ), and fix T > 0. Write σ(x) = xλ(x) 1/2. Then we set x y M(z, T ) Z T (x) = 2 S 0 S 0 σ 2 dz dy, (z) so that in particular, Z T (x) = 2σ2 (x)m(x, T ). And let: T G T (x, t) = F(T ) M(x, s) ds Z T (x) H T (x) = T R(x) T t [M(x, s) f (s)] ds + Z T (x)

Optimality Then there are two key results: Proposition For all (x, t, T ) R R + R + : G T (x, t) + H T (x) F(t) in D, G T (x, t) + H T (x) = F(t) in D C. Note that it follows that the reversed barrier stopping time attains equality. We call the inequality above a pathwise inequality

Optimality Define Q(x) = x S 0 y S 0 2σ(z) 2 dz dy Lemma Suppose f is bounded and for any T > 0, UI. Then for any T > 0, the process ( ) G T ( X t τd, t τ D ); 0 t T ( ) Q( X t ); 0 t T is is a martingale, and ( ) G T ( X t, t); 0 t T is a supermartingale. Note that we only have martingale properties up to T!

Optimality Theorem Suppose { τ D is Rost s } solution to the SEP, and for all T > 0, Q( X t ); 0 t T is a uniformly integrable family. Then τ D maximises EF(τ) over τ : X τ µ. This is just Chacon s optimality result. Note that the UI condition is easily checked when σ(x) = x.

Optimality: sketch of proof So for any solution τ to the Skorokhod embedding problem, (if we assume also τ, τ D T!) EG T ( X τ, τ) + EH T ( X τ ) EF(τ). But EH T ( X τ ) depends only on the law of X τ, and EG T ( X τ, τ) EG T ( X τd, τ D ) = G T ( X 0, 0). In addition, we get equality, G T ( X τd, τ D ) + H T ( X τd ) = F(τ D ), for the Rost stopping time, so EF(τ) EF(τ D ).

Optimality The additional T in the construction means that we cannot construct a pathwise inequality for all cases, even though we can prove optimality in general by a limiting argument. However the functions G T, H T, Z T can be interpreted in the limit provided we can find α > 1 such that for t large: C F (t) C O(t α ). In this case, we do indeed have a pathwise inequality for all t 0. In C. & Wang, we provided a similar proof of optimality for Root s embedding, where the dependence on T is no longer necessary, and the proof will go through in general.

Practical implementation Well-known results on viscosity solutions mean that we can use standard discretisation methods (Barles-Souganidis, c.f. Oberhauser & dos Reis) for PDEs to find u, and thus the reversed barrier. In fact, with a little extra work, we can even use implicit methods for Rost, this seems necessary (lots of detail at the start!) Can then compute the hedging strategies, and the upper and lower price bounds.

Numerical Implementation Payoff: F (t) = 2(t K ), K 0.2. Under the true model. 10 x 10-4 8 Intrinsic value Superhedge (Rost) Subhedge (Root) 6 4 Value 2 0-2 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Numerical Implementation Payoff: F (t) = 2(t K ), K 0.2. Under the incorrect model. 3.5 x 10-3 3 Intrinsic value Superhedge (Rost) Subhedge (Root) 2.5 2 Value 1.5 1 0.5 0-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Heston-Nandi model The Heston model is given (under the risk-neutral measure) by: ds t = rs t dt + v t S t db t, dv t = κ(θ v t ) dt + ξ v t d B t, where B t and B t are Brownian motions with correlation ρ. The Heston-Nandi model is the restricted case where ρ = 1, and so B t = B t. Note that v t = σt 2 in our previous notation, so we are interested in options on T 0 v t dt.

Heston-Nandi and Barrier stopping times Using Itô s Lemma, we know d(log(e rt S t )) = 1 2 v t dt + v t db t ( ( κθ κ = ξ ξ + 1 ) ) v t dt 1 2 ξ dv t. Solving, we see that ( e rt ) S T log S 0 = 1 ξ (v 0 v T ) + κθ ξ T ( κ ξ + 1 ) T v t dt. 2 0 Since v T is mean reverting, (v T v 0 ) (θ v 0 ) will be comparatively small for large T. In this case, we can write: T ( κ v t dt R T (e rt S T ) = ξ + 1 ) 1 [ ( )] κθ 2 ξ T + log S0 e rt S T 0

Heston-Nandi and barriers Samples from the Heston-Nandi model, and the corresponding barrier function. And an uncorrelated Heston model 3 2.8 Rost Root Simulation 2.6 2.4 2.2 Asset Price 2 1.8 1.6 1.4 1.2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Integrated Variance

Theorem Large T asymptotics Let M > 0 and suppose ξ, θ, κ, r > 0, ξ 2κ are given parameters of a Heston-Nandi model, Q HN. Suppose Q T is the class of models Q satisfying E QHN (S T K ) + = E Q (S T K ) + for all K 0. Then there exists a constant κ, depending only on M and the parameters of the Heston-Nandi model, such that for all convex, increasing functions F(t) with suitably smooth derivative f (t) = F (t) such that f (t), f (t) M, and for all T 0 E QHN F ( log S T ) Heston-Nandi is asymptotically optimal. inf E Q F ( log S T ) + κ. Q Q T

Sketch of Proof The function R T (x) is associated with a Root barrier. For this barrier, we can compute a martingale inequality that is, there exists a (sub-)martingale G and function H such that: F(v t ) G(e rt S t, v t ) + H(e rt S t ), t 0 and the Heston-Nandi model almost attains equality/martingality Can bound all approximations uniformly

Numerical Evidence The theorem is rather weak numerical evidence suggests there is more to be said: 0.25 Heston Rost Root 0.2 0.15 Option Price 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 Maturity

Conclusion Robust lower & upper bounds on variance options finding Root & Rost s barriers Can characterise (and compute) the barriers Explicit construction of robust super/sub-hedging strategies Heston-Nandi model is asymptotically extreme for variance options